1. Introduction
A relatively new steel–concrete composite beam using precast hollow core slabs (CBHCSs), as shown in Fig. 1(a), is proposed herein. Compared with previous composite beams, such new composite beams offer the following advantages: ① The CBHCSs can extend the structural span owing to the lighter mass of the hollow core slab (HCS); ② both the steel beam and precast HCS can be prefabricated in a shop, thereby obviating significant amounts of onsite wet work; ③ the integrity, strength, and stiffness are improved significantly, and water leakages are alleviated owing to the use of cast-in-place (CIP) concrete on the precast HCS; and ④ cracks on the floor surface are avoided owing to the mesh bars configured in the CIP concrete layer. This study focused only on the composite beam highlighted in the dashed area shown in Fig. 1(a). Therefore, only this region of the floor was represented in the experiments and finite element (FE) simulations.
Fig. 1. (a) Details of CBHCSs; (b, c) the monitoring system, including the force-measuring plate (ME-K3D160, Germany) and the data acquisition system (DH5922N, Jiangsu Donghua Testing Technology Co., Ltd., China); and (d) the field test.
Studies pertaining to shear behavior[
1–
2], flexural behavior [3– 9], thermal performance, and fire performance[
10,
11] have been conducted. Although the HCS is light and can stretch the structural span, its longer floor system is likely to experience vibrations from human activities[
12–
14]. In fact, the vibrational behavior of this relatively new composite beam system should be further investigated, as studies pertaining to it are scarce.
To alleviate the abovementioned vibration problem, the walking forces excited by humans must be elucidated. Harper [
15] performed walking tests, investigated the mechanics of walking, and proposed an M-shaped forcing function. Blanchard et al. [
16] defined the walking force function (vertical force
Fvertical and horizontal force
Flateral) using the Fourier series as follows:
where
G is the human weight (N);
i is the
ith harmonic;
t is the time;
αvertical i and
αlateral i are the vertical and horizontal coefficients of the Fourier series, respectively;
φvertical i and
φlateral i are the vertical and horizontal phase angles of the Fourier series, respectively;
is the stride rate; and
n is the total number of contributing harmonics.
The duration of single-step excitation
Tp is an important vibration parameter. It begins when the footfall touches the measuring force plate and ends when the footfall leaves the plate completely.
Tp reflects the walking pace. In previous studies[
17–
20], the stride rate
was used to determine the dynamic loading factors (DLFs) for the walking pace function using the Fourier series. However, few studies have been conducted to relate the DLFs with
Tp based on the simulation of a walking model.
The objectives of this study are as follows:
(1) To establish the Fourier series forcing function with Tp for one-step walking experimentally;
(2) To analyze the data measured from steady-state walking movements and establish the dynamic characteristics of the CBHCS;
(3) To conduct a sensitivity study using the finite element method;
(4) To derive and propose analytical expressions for the fundamental frequency and peak acceleration of a CBHCS system.
2. Measurement of walking forces
2.1. Experimental plan
The monitoring system comprised a measuring device (Fig. 1(b)) and a data acquisition system (Fig. 1(c)). The measuring device was a custom-designed force-measuring plate system with a total length of 3520 mm, including one force plate (500 mm long), two ramp wooden boards (1500 mm long each), and two 10 mm gaps between the force measuring plate and each ramp wooden board (Fig. 1(b)). The 10 mm gaps were designed to avoid undesired contact between the ramp wooden boards and the force plate. The force-measuring plate comprised one high-precision load sensor and two Q345 steel plates, as shown in Fig. 1(b). The data acquisition system (Fig. 1(c)) was used to record the signals transformed from the force-measuring plate system. To ensure that the foot is placed at the center of the force-measuring plate, a digital video camera was used to record each test and to identify whether the walking response would be useful for further vibration analysis. Therefore, only six among twelve traces in each walking test were selected to derive the walking force expression in this study.
Twenty-five volunteers participated in the walking force test. The test participants were instructed to walk through the forcemeasuring plate system (Fig. 1(d)) from a marked starting line. Each test participant walked forward and backward six times.
2.2. Walking force model
The typical load–time history under the walking excitation by Participant 1 is shown in Fig. 2(a). The one-step walking curve shown in Fig. 2(b) is similar to that of Harper’s study [
15]. Because the vertical vibration primarily controls the building floor’s vibration serviceability, only the walking force in the direction of gravity is discussed herein.
The peak walking force (Fmax) is associated with the walking posture, stride rate, and human weight in particular. A total of 150 traces (6 traces/participant × 25 participants) were acquired and used to calculate the ratios of the peak walking force (Fmax) to the human weight (G). The ratios varied from 1.06 to 1.56, in which more than 85% of them were between 1.10 and 1.35, and the mean ratio (MEAN) was 1.2267 with a standard deviation (STD) of 0.1027 and a coefficient of variation (CV) of 0.0837, as shown in Fig. 2(c).
Additionally, Tp (Fig. 2(b)) which begins when the footfall touches the force-measuring plate and ends when the footfall leaves the force measuring plate completely, is another important vibration parameter. Fig. 2(d) shows the relationship between the ratio of the peak walking force to the human weight (Fmax/G) and Tp. Moreover, more than 80% of the data were within the 15% error range (Fig. 2(d)), indicating a satisfactory distribution. Hence, a mean Fmax/G of 1.2267 and a Tp of 0.6236 s were suggested for the walking vibration.
Fig. 2. Test results under the walking excitation. (a) Typical load–time history and (b) typical one-step curve of Participant 1; (c) distribution of the ratios of peak walking force (Fmax) to the human weight (G); (d) relationship between Fmax/G and Tp. MEAN: the mean ratio of the peak walking force (Fmax) to the human weight (G); STD: the standard deviation for the ratio of the peak walking force (Fmax) to the human weight (G); CV: the coefficient of variation for the ratio of the peak walking force (Fmax) to the human weight (G).
When conducting the theoretical analysis of the floor dynamic response due to walking excitation, G and Tp were considered; subsequently, the analysis was used to develop the walk forcing function. The following Fourier series was used to define the onestep walk forcing function F(t):
where n is the total number of contributing harmonics, αn is the DLF, and φn is the phase angle. An and Bn are the Fourier coefficients.
Ebrahimpour and Hamam [
21] stated that for walking activities, the higher-order harmonics (i.e.,
n = 2,
n = 3, ...) were insignificant, particularly as the number of participants increased. As
n increased, the simulation accuracy of the formulation increased, but more computations were incurred. Various orders of harmonics (
n = 1–4) were considered in the formulation analysis, and the typical force–time histories from Participant 1 are shown in Fig. 3(a).
In comparison with the test results, Eq. (3) shows a better accuracy with n = 2, which yields a maximum error of 0.01% to –6.04%. Therefore, F(t) can be modified as follows:
Based on the 150 walking tests, relationships between the DLFs (α0, α1, and α2) and Tp, and those between the phase angles (φ1 and φ2) and Tp (based on Eq. (9)) can be established using the leastsquares method. It is noteworthy that Tp ranged from 0.52 to 0.87 s. The main findings are as follows:
(1) As Tp increased, the coefficient α0 decreased linearly, namely, α0 = –0.2775Tp + 0.9799, with an average α0 of 0.790.
(2) As Tp increased, the coefficient α1 increased linearly, that is, α1 = 0.9246Tp –0.4192, with an average α1 of 0.214.
(3) As Tp increased, the coefficient α2 decreased linearly, that is, α2 = 0.4616Tp + 0.6987, with an average α2 of 0.382.
(4) The linear correlation between the phase angle φ1 (or φ2) and Tp was unclear. However, the values of the phase angles (φ1 and –0.5π) were primarily in the range of –0.5π to –0.3π.
(5) During the Fourier series expansion, the effect of the phase angle on the vibration periods was insignificant. However, the change in phase difference
(=
φ1 –
φ2 ) was important for determining the walking model. The value of
was predominantly 0, which may be assumed when defining the walking model. Hence, both
φ1 and
φ2 can be set as –0.5π for further analysis, which coincides with the values proposed by Chen et al. [
19].
After determining the relationships among the force function parameters (i.e., the DLFs and phase angles in Eq. (9)) and Tp, the force–time responses were simulated using Eq. (9) and then plotted and compared with the test results. Typical fitted force–time curves were compared, as shown in Fig. 3(b). In all 150 groups of data, the maximum errors between the simulated and test results were less than 15%, with most of them being less than 10%. This suggests that Eq. (9) can be used to reasonably simulate the onestep walking force.
Fig. 3. Comparison of the walking force model (by Participant 1). (a) Comparisons between the measured and calculated curves for various orders of harmonics (n = 1–4) under the walking excitation; (b) comparisons of the fitted and test curves (n = 2).
3. Vibration tests of CBHCS under walking
3.1. Test specimens
Seven full-scale tests were conducted on CBHCSs under walking. As indicated in Table 1, the beam span was either 6.0 m (four specimens) or 8.4 m (three specimens). The following two typical sizes of the H-shape steel beam were selected (depth × flange width × web thickness × flange thickness): 300 mm × 150 mm × 6.5 mm × 9 mm and 400 mm × 200 mm × 8.0 mm × 13 mm. Each H-shaped steel beam comprised two rows of headed shear studs measuring 19 mm (diameter) × 120 mm (height) prewelded on its top flange (one row on each side of the flange). The transverse spacing of the studs was 50 mm, whereas different longitudinal spacings of 175, 280, and 475 mm were considered to reflect 100%, 70%, and 30% degrees of shear connection [
22], respectively. The precast HCSs were placed on the top flanges of steel beams with a 25 mm bearing length, as shown in Fig. 1(a). Two floor widths of 2100 and 1800 mm were designed to investigate the effect of the floor width on the vibrational behavior of the CBHCS. To avoid concrete splitting between the headed shear studs (typically 19 mm in diameter and 120 mm long) and precast HCS (150 mm height), transverse tie reinforcements measuring 18 mm (diameter) × 1000 mm (total length of 450 mm in each HCS plus 100 mm gap) were inserted in the open cores of the HCSs at every 190 mm. To enhance the integrity of the composite beam and minimize walking noise, CIP concrete topping (60 mm thick) was added to the HCS. The gap between the HCSs was filled with
in-situ concrete. In addition, mesh bars (diameter, 6 mm; spacing, 200 mm) were placed in the CIP concrete layer to avoid concrete cracking. The thickness of the CIP concrete layer was 60 mm. The complete details of the test specimens are presented in Table 1.
Table 1 Details of the CBHCS specimens and test participants.
Steel beam type: depth (mm) × flange width (mm) × web thickness (mm) × flange thickness (mm).
3.2. Test setup and instruments
Figs. 4(a)–(c) show schematic illustrations of the locations of accelerometers used for obtaining measurements from the transducers. The spacing between two adjacent accelerometers is clearly indicated for each direction. The monitoring system [
23] comprised accelerometers (Type 2D001V, Jiangsu Donghua Testing Technology Co., Ltd., China) with an acceleration range of 2
g and a data acquisition system (Model DH5922N, Jiangsu Donghua Testing Technology Co., Ltd.) (Figs. 4(d) and 1(c)).
Fig. 4. Measurement locations and accelerometer numbering for (a) CBHCSs 1, 5, and 6, (b) CBHCS 7, and (c) CBHCSs 2, 3, and 4 (dimensions in mm); (d) the accelerometer (2D001V); and (e) the photo of walking test (U1 is perpendicular to U2 in horizontal direction, U2 is the horizontal walking direction, and U3 is the vertical direction).
This study focused on the vibration behavior of CBHCSs caused by walking impacts. For comparison, various walking rates were considered for the seven specimens (Table 1). Walking path A06– A07–A08–A09–A10 (Figs. 4(a)–(c)) was selected for each slab panel. Seven test participants were instructed to walk back and forth on each CBHCS for at least 60 s at the respective normal speed (Table 1), as shown in Fig. 4(e).
3.3. Results and discussion
3.3.1. Acceleration in time domain
Typical acceleration traces at location A08 measured during the walking tests are shown in Fig. 5. All peak accelerations under walking excitation on each specimen are listed in Table 2. To quantify the human response to floor vibrations, ISO 2631-2 [
24] lists the acceleration limits for mechanical vibrations, which are expressed in multiples of basic root–mean–square (RMS) accelerations. The RMS acceleration (
) can be calculated as follows:
where
N is the number of data points considered during the integration period and
represent the individual accelerations. In this study,
n was set to 1000 for an integration period of 1 s.
The peak RMS acceleration is referred to as the maximum transient vibration value (MTVV), which can be calculated as follows:
The average RMS acceleration (
) can be calculated as follows:
The peak and RMS accelerations at mid-span under walking impact are listed in Table 2, where
χrp is the coefficient used to describe the relationship between the MTVV and peak acceleration, that is,
The following are indicated in Table 2:
(1) No frequency weighting was used in this study. The degree of shear connection (100%, 70%, or 30%) did not significantly affect the vibrational frequency. Therefore, the stud connection may be disregarded when evaluating the vibrational serviceability of CBHCSs.
(2) The span and floor width of the specimen affected the vibrational behavior of the CBHCSs. Therefore, these parameters should be considered in the vibration evaluation.
(3)
χrp varied between 0.55 to 0.78. It is noteworthy that the average
χrp coefficient was 0.58 for the CBHCSs under walking impact, calculated based on Grubbs’ criterion [
25].
Fig. 5. Typical measured and root–mean–square (RMS) acceleration responses (Location A08 of CBHCS-7).
3.3.2. Experimentally determined mode shapes
The mode shapes of the CBHCS floor system were extracted using the enhanced frequency domain decomposition (EFDD) method [
26]. The first two mode shapes obtained from the experiments are shown in Fig. 6, where the first and second modes represent the typical bending and second-order bending shapes, respectively.
The representative experimentally obtained natural frequencies and damping ratios are listed in Table 2. The fundamental frequency for the 6.0 m-span specimens was approximately 16 Hz, which decreased to approximately 11.5 Hz when the span increased to 8.4 m. A damping ratio of 1% was suggested for further analysis.
Table 2 Testing result comparison.
: the maximum transient vibration value;
: the peak acceleration;
χrp: the coefficient used to describe the relationship between
and
(i.e.,
).
Fig. 6. Representative experimentally obtained mode shapes. (a, b) First mode shape (16.07 Hz) and second mode shape (52.69 Hz) of Specimen CBHCS-1 (6.0 mspan); (c, d) first mode shape (11.48 Hz) and second mode shape (38.66 Hz) of Specimen CBHCS-2 (8.4 m-span).
4. Finite element analysis
4.1. Model details
The natural frequencies and mode shapes of the CBHCS were predicted using the general FE program ABAQUS [
27]. The element of continuum stress/displacement three-dimensional with eightnode by reduced integration (C3D8R), the shell element with four-node by reduced integration (S4R), and the two-dimensional truss element of three-node piezoelectric (T3D2) were used to represent the concrete, steel beam, and reinforcements (or studs), respectively [
28]. All material properties were determined based on the appropriate material tests. The elastic constants (elastic modulus of 3 ×10
4 MPa and Poisson’s ratio of 0.2 for concrete; elastic modulus of 2 ×10
5 MPa and Poisson’s ratio of 0.3 for steel) and density (7850 kg·m
–3 for steel and 2400 kg·m
–3 for concrete) were considered. A damping ratio of 1% was used in this study. The following boundary conditions were assumed to represent the simple support at each beam end: U1 = U2 = U3 = UR2 = UR3 = 0 and U1 = U3 = UR2 = UR3 = 0 (Fig. 4(e)). U1 = 0 means the translational degree of freedom (DOF) in the
x-axis is 0 and UR1 = 0 means the rotational DOF about the
x-axis is 0.
In the FE model, the floor was regarded as a solid instead of the HCS for simplicity and efficiency when using the S4R elements to model the floor. The het in Table 3 is the equivalent thickness of the solid floor with the same section inertia and mass as those of the CBHCS. The relationships between the het of the solid floor and the overall thickness h of the original CBHCS section are listed in Fig. 7 and Table 3.
Fig. 7. Transformed Area Method. hHCS: the thickness of HCS floor; hcast: the thickness of cast in-situ floor; Lmajor: length of major axis of elipse; Lminor: length of minor axis of elipse; Lspacing: the hole spacing of CBHCS.
Table 3 Relationships between the equivalent thickness het of the solid floor and thickness h of the original CBHCS section.
4.2. Model validation
As shown in Fig. 8(a), the natural frequencies for the first two modes were similar between the FE analysis and test, with a relative error of less than 10% (Table 4). The following modal assurance criterion (MAC) [
29] was used to identify the matching modes:
, where Φ
r is the identified testing mode (mode
r) and
is the calculated mode (mode
s). The MAC takes value between 0 (representing no consistent correspondence) and 1 (representing a consistent correspondence). MAC values larger than 0.9 indicate consistent correspondence whereas small values indicate poor resemblance of the two mode shapes. The MAC values for the first two modes generally exceed 0.95 (Table 4), indicating the high accuracy of the mode shapes from FE analysis.
A comparison of time histories is presented in Fig. 8(b), which shows that the relative error was less than 15% for peak accelerations and less than 10% for MTVVs. In summary, the FE model was validated and can be used for parametric studies.
Fig. 8. Model validation. (a) Comparison of natural frequencies between the FE analysis and test; (b) comparison of acceleration histories between the FE analysis and test (CBHCS-1). L: the beam span.
Table 4 Comparison of natural frequencies and modal assurance criterion values (MACs).
4.3. Sensitivity study
Based on the test and FE analysis results, the natural frequencies and mode shapes of the CBHCSs with different shear connection degrees were similar for the first two modes. Hence, Specimen CBHCS-2 (
L = 8.4 m, the tested acceleration
) was selected to conduct the parametric study. A total of 20 FE analyses were conducted by varying one parameter at each time to further investigate the effects of different factors on the vibrational behavior of the CBHCSs. The walking frequency was set to 2.0 Hz. The primary factors included the floor thickness, steel beam type, contact time, and human weight. The FE analyses are summarized in Table 5 and Fig. 9.
Table 5 Summary of the FE analyses.
Taking CBHCS-205-400-0.72-610 as an example to explain the designation: 205 is the equivalent floor thickness in mm; 400 indicates the steel beam HN 400 × 200 × 8 × 13 having a depth of 400 mm; 0.72 is the contact time in second; and 610 is the weight of test participant in N.
The main FE analysis results are discussed as follows.
4.3.1. Floor thickness
The effect of the het on the natural frequency and acceleration response was investigated by varying the het values (150– 205 mm), as shown in Tables 3 and 5. The calculated fundamental and second-order frequencies increased slowly with het as the bending stiffness increased (Fig. 9(a)). In addition, the peak acceleration, MTVV, and the average RMS (ARMS) increased with het, particularly when het > 195 mm (Fig. 9(b)). Because the opening ratio of the HCS increases with het, local vibrations will occur when the opening ratio of the HCS is increased; this will result in greater vibrational responses, which should be focused upon in vibration analysis.
Fig. 9. Parametric analysis results of the CBHCS. Effects of (a, b) floor thickness, (c, d) steel beam type, (e, f) contact time, and (g, h) human weight on frequency and acceleration, respectively.
is the acceleration value calculated from the FE method;
is the measured acceleration value;
hbeam: the steel beam height; ARMS: the average RMS.
4.3.2. Steel beam type
The effect of the steel beam type on the natural frequency and acceleration response was investigated by changing the arrangement of the steel beam, as shown in Table 5. Four different steel beam arrangements were investigated and compared. The calculated fundamental and second-order frequencies increased with the steel beam height (hbeam) because the bending stiffness increased (Fig. 9(c)). Fig. 9(d) shows that the peak acceleration, MTVV, and ARMS were affected significantly when hbeam ≤ 400 mm, whereas these values decreased significantly when hbeam > 400 mm owing to the increasing mass and stiffness of the specimen.
4.3.3. Contact time TP
The effects of TP on the natural frequency and acceleration response were investigated, as shown in Table 5. The natural frequencies of the first two modes were barely affected because TP contributed insignificantly to the floor stiffness and mass (Fig. 9(e)). TP did not affect the stiffness or mass of the structure, as the applied walking forces using TP did not interfere with the floor natural frequencies. When TP increased, the peak acceleration, MTVV, and ARMS decreased owing to the decreasing impact force (Fig. 9(f)).
4.3.4. Human weight
The effect of human weight (505, 610, 681, 812, and 900 N) on the natural frequency and acceleration response was investigated, as shown in Table 5. The natural frequencies of the first two vibration modes were barely affected because G contributed insignificantly to the floor stiffness and mass (Fig. 9(g)). G did not affect the stiffness or mass of the structure, as the applied walking forces using G did not interfere with the floor natural frequencies. Fig. 9(h) shows that as G increased, the peak acceleration, MTVV, and ARMS increased with the impact force.
5. Evaluation of vibration serviceability of CBHCS
5.1. Frequency
A simply supported beam subjected to dynamic loading is shown in Fig. 10(a), where the important beam properties are the flexural stiffness
EI (
E is the elastic modulus;
I is the area moment of inertia of the beam cross-section about the axis of interest) and the mass per unit length
, both assumed to be constant along the span
L. The transverse loading force
F(
x,
t) varies with position and time, resulting in various transversedisplacement responses
.
Fig. 10. Theory analysis models. (a) Simple beam subjected to dynamic loading; (b) a single-degree of freedom (SDOF) system.
For free vibrations, that is, F(x,t) = 0, the equation becomes
where iv in the superscripts means the fourth derivative.
means the time related displacement responses.
means the position related displacement responses.
If
and the following boundary conditions are imposed
where M(0) or M(L) Þmeans the bending moment at x = 0 or x = L.
Then
where Ω is the coefficient.
Because
= L 0,
Ω,
Therefore, the natural frequencies can be calculated as follows:
where ωn is the circular frequency.
Using the theory above, the theoretical fundamental frequency (n = 1) of the specimens can be obtained. Table 6 shows a comparison between the theoretical and test results. As shown, the relative errors were generally less than 7%. Hence, the fundamental frequency, calculated using Eq. (18), was considered acceptable.
Table 6 Comparison of the fundamental frequencies and accelerations.
5.2. Acceleration
The equation of motion of a single-degree-of-freedom (SDOF) system can be reduced to the following form:
where
t is the time,
m is the mass of vibration,
c the viscous damping coefficient,
k the stiffness,
v(
t) the displacement of the mass, and
F(
t) the force applied to the mass. In this study, it is assumed that the system shown in Fig. 10(b) is subjected to a harmonically varying load
F(
t) of amplitude
F0 and circular frequency
In this case, the differential equation of motion becomes
As
(
β is the damping ratio) and
the following is obtained:
The solution of Eq. (21) can be expressed as
where the amplitude
A =
the oscillatory circular frequency
the magnification factor
and the circular frequency ratio
In Eq. (22), the first term
represents the transient response, which vanishes rapidly with damping and is negligible, thereby yielding an acceleration expressed as follows:
The peak acceleration is expressed as
where
and
The AISC Design Guide 11 [
30] recommends 700 N for the human weight
G.
The coefficient α is calculated as follows:
can be re-expressed as
where
m =
and the boundary coefficient
ρ = 0.5 represent simple supports. In this study, the human mass was assumed to be 65 kg (i.e.,
G = 650 N), based on the walking force measurements discussed in Section 2. The calculated peak accelerations based on the equation of motion for an SDOF system were compared with those from the tests, and a relative error less than 50% was indicated (Table 6). Acceptable results are achievable if the actual walking force is employed to calculate the acceleration response of each specimen using the FE method. However, it is unrealistic to perform further floor vibration evaluations using the actual walking force each time. A unified walking force (i.e., coefficient
α ×
G) is typically used to obtain the peak acceleration. Therefore, an error less than 50% (Table 6) is considered acceptable.
The relationship between the MTVV and peak acceleration can be expressed as
The average
χrp coefficient was calculated to be 0.58, compared with 0.2 reported by Zhou et al. [
31]. As shown in Table 6, the relative error of the calculated MTVVs was 6% on average and 41% at the maximum. Better results can be obtained if the actual walking forces are employed to calculate the acceleration response of each specimen. In general, the applied walking force should be calculated using Eq. (9). Therefore, the calculated force cannot truly reflect the walking posture or the actual vibration response. Furthermore, the thresholds of floor vibration specified in AISC Design Guide 11 [
30] are 50, 150, and 500 mm·s
–2 for different types of floors. This implies that the order of magnitude for accelerations, instead of the actual acceleration value, is typically used in floor vibration evaluations. Hence, the MTVVs calculated using Eq. (27) were considered acceptable.
6. Conclusions
In this study, 150 load–time histories from walking were obtained using a force measuring plate. Vibration tests on seven CBHCS specimens were conducted in addition to theoretical analysis. This study focused on the behavior in the vicinity of the steel beam and the vibratory motion of a steel–concrete CBHCS. On a real floor, vibrations occur at the center of the slab and should be addressed in future studies. The following conclusions were obtained from this study:
(1) The ratio of peak walking force to human weight ranged from 1.10 to 1.35. A Fourier series walking function with n = 2 was derived, that is, Eq. (9). Relationships between the function parameters (DLFs and phase angles) with Tp were determined as follows: α0 = –0.2775Tp + 0.9799, α1 = 0.9246Tp –0.4192, and α2 = –0.4616Tp + 0.6987; and φ1 = φ2 = –π/2. A mean Fmax/G ratio of 1.2267 and Tp of 0.6236 s were suggested for the walking vibration on the CBHCS floor.
(2) The CBHCS floor system indicated a high frequency (> 10 Hz) and low damping (~1%). The mode shapes predicted from the FE analysis matched those from the tests. The CBHCS floor exhibited flexural mode shapes during human walking.
(3) FE analyses indicated that the primary factors affecting the vibration of the CBHCS floor were the floor thickness, steel beam type, contact time (Tp), and human weight (G). However, Tp and G barely affected the natural frequencies of the first two vibration modes because they contributed insignificantly to the floor stiffness and mass. This is because Tp and G did not affect the stiffness or mass of the structure, as they were applied forces that did not interfere with the natural frequencies.
(4) The theoretical fundamental frequency of the CBHCS floor can be obtained using beam vibration theory. The relative errors were generally less than 6%, indicating that the fundamental frequency calculated using Eq. (18) was reasonably accurate.
(5) The calculated peak accelerations based on the equation of motion for an SDOF system were compared with those from the tests, and a relative error less than 50% was indicated. An average χrp coefficient of 0.58 was suggested to obtain the MTVV. The MTVV result (maximum error less than 50%) is acceptable because the order of magnitude of the accelerations instead of the actual acceleration value is typically used in floor vibration evaluations.
Acknowledgment
The authors acknowledge the financial support provided by the National Natural Science Foundation of China (51890902 and 51708058).
Compliance with ethics guidelines
Jiepeng Liu, Shu Huang, Jiang Li, and Y. Frank Chen declare that they have no conflict of interest or financial conflicts to disclose.
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